Feller gives in Volume I, "An Introduction to Probability Theory and It's
Applications", that if one starts at the origin and moves at random the
probability of returning to the origin in one or two dimensions is one
and
in three dimensions it is about .35. (In two dimensions there are 4
random
choices and in 3 dimensions, 6.)

Question: In 3 dimensions if you start at the point (10000,10000,10000)
do
you arrive at a different answer?

Does your question refer to returning to the starting point or
returning to the origin?

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Hi Barry,

It concerns "returning" (getting there the first time) to the origin.

Actually, I suspect that, in the three dimensional case, the starting point
influences the probability of "returning" to the origin. I'm wondering what
the Group thinks (knows).

Best wishes, Jim

I can recommend this (free) monograph by Doyle and Snell on the
subject:

In two dimensions, a random walk on the "graph" connecting
the usual integer lattice points visits every point in the plane
with probability 1. So in that setting it doesn't matter whether
we ask about "returning" to the origin or going from one point
to another, the probability is the same. This can be deduced
from the divergence of SUM 1/n.

In three dimensions, as Jim quotes Feller, the chance of
returning to an original starting point is positive and strictly
less than 1 (about .35). This is related to the convergence
of SUM 1/(n^2), although the connection is not elegant when
made rigorous. The search for a simple connection motivates
much of the monograph above.

In short we can define a function that gives the probability
that a random walk starting at the origin will visit (at some
subsequent step) any specific point in the integer lattice.
Evaluation of this function at (1000,1000,1000) will give by
symmetry the answer to Jim's question, and in general
this function is monotone decreasing as we go farther
from the origin.

Feller gives in Volume I, "An Introduction to Probability Theory and
It's
Applications", that if one starts at the origin and moves at random
the
probability of returning to the origin in one or two dimensions is
one
and
in three dimensions it is about .35. (In two dimensions there are 4
random
choices and in 3 dimensions, 6.)

Question: In 3 dimensions if you start at the point
(10000,10000,10000)
do
you arrive at a different answer?

Does your question refer to returning to the starting point or
returning to the origin?

Remove del for email

Hi Barry,

It concerns "returning" (getting there the first time) to the origin.

Actually, I suspect that, in the three dimensional case, the starting
point
influences the probability of "returning" to the origin. I'm wondering
what
the Group thinks (knows).

Best wishes, Jim

I can recommend this (free) monograph by Doyle and Snell on the
subject:

In two dimensions, a random walk on the "graph" connecting
the usual integer lattice points visits every point in the plane
with probability 1. So in that setting it doesn't matter whether
we ask about "returning" to the origin or going from one point
to another, the probability is the same. This can be deduced
from the divergence of SUM 1/n.

In three dimensions, as Jim quotes Feller, the chance of
returning to an original starting point is positive and strictly
less than 1 (about .35). This is related to the convergence
of SUM 1/(n^2), although the connection is not elegant when
made rigorous. The search for a simple connection motivates
much of the monograph above.

In short we can define a function that gives the probability
that a random walk starting at the origin will visit (at some
subsequent step) any specific point in the integer lattice.
Evaluation of this function at (1000,1000,1000) will give by
symmetry the answer to Jim's question, and in general
this function is monotone decreasing as we go farther
from the origin.

regards, chip

Hi Chip,

Thank you. I suspected as much in that if you start at the origin you have
a 1/6 probability of reaching the origin on the second move. It's
interesting to learn that, as you move the starting point away from the
origin, the probability of reaching the origin diminishes. A little
reflection does indicate this as a logical conclusion, but seemingly logical
conclusions often blow up in your face.

In short we can define a function that gives the probability
that a random walk starting at the origin will visit (at some
subsequent step) any specific point in the integer lattice.
Evaluation of this function at (1000,1000,1000) will give by
symmetry the answer to Jim's question, and in general
this function is monotone decreasing as we go farther
from the origin.

THe probability that a random walk starting at the origin will
eventually return to the origin should be the same as the probability
that a random walk starting at (1,0,0), (0,1,0), (0,0,1), (-1,0,0),
(0,-1,0), or (0,0,-1) will eventually reach the origin.